Poisson Distribution Calculator

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate. It assumes events occur independently and at a constant average rate λ. Common examples include call center arrivals, defects in manufacturing, or website visits per minute.

A Poisson experiment is a statistical experiment with the following properties: the outcomes can be classified as successes or failures, the average number of successes (λ) in a defined region is known, the probability of success is proportional to the size of the region, and the probability of two successes in an infinitely small region is effectively zero. Examples include counting emails received per hour or accidents per year on a stretch of road.

Lambda (λ) is the expected number of events (successes) in a fixed interval — whether that interval represents time, area, volume, or any other measurable region. For example, if a call center receives an average of 5 calls per minute, then λ = 5. Lambda must always be a positive number.

A Poisson random variable (x) is the actual number of successes or events you want to evaluate the probability for. It must be a non-negative integer (0, 1, 2, 3, …). For instance, if you want to know the probability of exactly 3 calls arriving in a minute when the average is 5, then x = 3.

A cumulative Poisson probability is the probability that the Poisson random variable falls within a specified range. For example, P(X ≤ x) sums all probabilities from 0 up to x — this is the cumulative distribution function (CDF). Similarly, P(X < x), P(X > x), and P(X ≥ x) are other cumulative forms computed by this calculator.

For a Poisson distribution, the standard deviation equals the square root of λ: σ = √λ. This is a unique property — the mean and variance are both equal to λ, so the distribution is fully described by a single parameter. A larger λ means both a higher expected count and more spread in the distribution.

The Poisson probability mass function is: P(X = x) = (e^−λ × λˣ) / x!, where e ≈ 2.71828 is Euler's number, λ is the average rate, and x is the number of events. The cumulative probabilities are found by summing the PMF over the relevant range of integer values from 0 up to x.

Use the Poisson distribution when the number of trials is very large and the probability of success on any individual trial is very small, so that the average rate λ = n × p remains moderate. In practice, Poisson is preferred when n > 20 and p < 0.05. When both n is small and p is not near zero, the binomial distribution is more appropriate.